Every Meromorphic Function is the Gauss Map of a Conformal Minimal Surface

Let M be an open Riemann surface. We prove that every meromorphic function on M is the complex Gauss map of a conformal minimal immersion M. R3 which may furthermore be chosen as the real part of a holomorphic null curve M. C3. Analogous results are proved for conformalminimal immersions M. Rn for any n > 3. We also show that every conformal minimal immersion M. Rn is isotopic through conformal minimal immersions M. Rn to a flat one, and we identify the path connected components of the space of all conformal minimal immersions M. Rn for any n >= 3.